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I am trying to think of this problem for quite some time. Let's say, we have two sets of wave functions $\lbrace|\psi\rangle\rbrace$ and $\lbrace|\phi \rangle\rbrace$ and they belong to two different Hilbert spaces. That is,

$$\hat{H_1}|\psi\rangle=E_1|\psi\rangle$$

and

$$\hat{H_2}|\phi\rangle=E_2|\phi\rangle.$$

In the real space $\bf{R}$, their functional domains are disjoint. That is, if $\psi(x)$ is defined in $x\le0$, $\phi(x)$ is defined in $x>0$.

In this case, is it possible to conceive some kind of superposition between the two waves? If so, how? I mean how do we define the superposed wave function and what can be said about the energy? This paper introduces such a concept

http://dx.doi.org/10.1119/1.18854

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  • $\begingroup$ In quantum field theory, there is a field in spacetime. But in nonrelativistic quantum mechanics, a wavefunction is not a function from $\mathbb R^3$ into the complex numbers, a wave function is a function from the configuration space of the system into the joint spin state of the system. $\endgroup$ – Timaeus Sep 26 '15 at 17:34
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    $\begingroup$ Where in the cited paper does it mention having two distinct Hilbert Spaces? It's a method of images solution to reflection off an infinite potential barrier. $\endgroup$ – Timaeus Sep 27 '15 at 1:25
  • $\begingroup$ @Timaeus Isn't the wave function in quantum mechanics defined as $f_{\psi}\colon x\to \langle x|\psi\rangle$, and thus exactly a function from $\mathbb{R}$ to the complex numbers? $\endgroup$ – gented Sep 27 '15 at 18:55
  • $\begingroup$ @GennaroTedesco Most definitely not. It is a function from configuration space into a joint spin state. The joint spin state is a tensor product of single particle spin states (one for each particle) and configuration space is $\mathbb R^{3n}$ when there are $n$ particles. $\endgroup$ – Timaeus Sep 27 '15 at 21:06
  • $\begingroup$ How would you then define the wave function for the free spinless particle, according to that definition? Can you address me to some literature to check that? I have somehow never encountered such definition before. $\endgroup$ – gented Sep 27 '15 at 21:12
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Two different Hibert spaces correspond to two different physical systems. Superposition of wave functions makes sence for one system (for one Hilbert space), since addition of vectors (quantum states) is defined in a particular vector space (Hilbert space). What you can do is to create a new Hilbert space by forming the tensor product of the two Hilbert spaces. And if you work in the coordinates representation, then you should expand the domain of the each wave function to the entire real line, by multiplying $\psi \left( x \right)$ whith the characteristic function of $\left( -\infty ,0 \right]$ and $\varphi \left( x \right)$ with the characteristic function of $\left( 0,+\infty \right)$.

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  • $\begingroup$ In such a case, what will happen to the inner product of the new Hilbert space ! $\endgroup$ – user35952 Sep 26 '15 at 17:22
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    $\begingroup$ If by saying "what will happen?" you mean "how is it defined?", then check this article: en.wikipedia.org/wiki/Tensor_product_of_Hilbert_spaces $\endgroup$ – user3257624 Sep 26 '15 at 17:26
  • $\begingroup$ Thank you very much for the reply. I shall go through this concept. One query: in the extended Hilbert space, does the two wave functions add normally? or do we need to construct some direct product or direct sum? $\endgroup$ – kolahalb Sep 26 '15 at 18:15
  • $\begingroup$ In the tensor product space, the states are products of the wave functions; what is added there is such products of them and not the individual wave functions $\endgroup$ – user3257624 Sep 26 '15 at 18:21
  • $\begingroup$ you are very welcome! $\endgroup$ – user3257624 Sep 26 '15 at 18:29

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